Determine where the function f has a derivative, as a function of a complex variable:
f(x +iy) = 1/(x+i3y)
I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?
Alright, so doing the partials and equating I get
x = 3y and
xy = 3xy
So, subbing first equation into the second we get,
3y^2=9y^2, so y=0. If y=0 then x=0. So differentiable only at 0+0i.
But, because 0 + i0 is undefined in the original function, would this be differentiable nowhere?
Am I approaching this right?
so just to clarify, in general there are cases where cauchy-riemann are not satisfied, however derivatives do exisit at some points? (In Other Words, if Cauchy-Riemann is not satisfied, it does not neccessarily mean the function isnt differentiable everywhere?)